Abstract
The concept of a weak factorization system for modules has found an application in one of the proofs of the celebrated flat cover conjecture for modules over a ring. We examine this notion in the context of S-acts over a monoid S. Bailey and Renshaw constructed a weak factorization system related to the existence of precovers of S-acts and showed that the class of flat right S-acts satisfies a related property they call saturated. In this paper we provide a factorization system, namely a weak factorization system with the unique mapping property, which is related to the existence of covers of S-acts with the unique mapping property. Moreover, we show that in this case, a class of monomorphisms associated with principally weakly flat right S-acts is saturated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahsan, J., Liu, Z.: A Homological Approach to the Theory of Monoids. Science Press, Beijing (2008)
Bailey, A., Renshaw, J.: Weak factorization systems for \(S\)-acts. Semigroup Forum 89, 52–67 (2014)
Bailey, A., Renshaw, J.: Covers of acts over monoids and pure epimorphisms. Proc. Edinb. Math. Soc. 57, 589–617 (2014)
Bailey, A., Renshaw, J.: Cover of acts over monoids II. Semigroup Forum 87, 257–274 (2013)
Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs. Oxford University Press (1995)
Kilp, M., Knauer, U., Mikhalev, A.: Monoids. Acts and Categories. Walter de Gruyter, Berlin New York (2000)
Mahmoudi, M., Renshaw, J.: On covers of cyclic acts over monoids. Semigroup Forum 77, 325–338 (2008)
Rasouli, H.: Categorical properties of regular monomorphisms of \(S\)-posets. Eur. J. Pure Appl. Math. 7, 166–178 (2014)
Renshaw, J.: Subsemigroups of free products of semigroups. Proc. Edinb. Math. Soc. 34, 337–357 (1991)
Renshaw, J.: Extension and amalgamation in monoids and semigroups. Proc. Lond. Math. Soc. 52, 119–141 (1986)
Renshaw, J.: Flatness and amalgamation in monoids. J. Lond. Math. Soc. 33, 73–88 (1986)
Renshaw, J.: Stability and flatness in acts over monoids. Colloq. Math. 92, 267–293 (2002)
Rosický, J., Tholen, W.: Factorization, fibration and torsion. J. Homotopy Relat. Struct. 2, 295–314 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Victoria Gould.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, H., Zhu, R. On factorization systems for S-acts. Semigroup Forum 107, 478–490 (2023). https://doi.org/10.1007/s00233-023-10377-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-023-10377-8